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arXiv:quant-ph/0208074 · Physical Review A

Two roles of relativistic spin operators

Daniel R. Terno

Published Jan 29, 2003• 65 citations
arXivHTMLPDF

Abstract

Operators that are associated with several important quantities, such as angular momentum, play a double role: they are both generators of the symmetry group and ``observables.'' The analysis of different splittings of angular momentum into ``spin'' and ``orbital'' parts reveals the difference between these two roles. We also discuss a relation of different choices of spin observables to the violation of Bell inequalities.

DOI
10.1103/physreva.67.014102

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Operators that are associated with several important quantities, such as angular momentum, play a double role: they are both generators of the symmetry group and observables.'' The analysis of different splittings of angular momentum into spin'' and ``orbital'' parts reveals the difference between these two roles. We also discuss a relation of different choices of spin observables to the violation of Bell inequalities.
Spin-s systems Angular momentum is one of the fundamental quantities of Newtonian physics, and in quantum physics its importance is at least as great. In quantum mechanics we often distinguish between two types of angular momentum: orbital angular momentum , which a system of particles possesses due to particle motion through space; and spin angular momentum , which is an intrinsic property of a particle. The distinction will be important later, but for now we will ignore it. We will here refer to angular momentum of any sort as “spin” and develop general-purpose mathematical tools for its description. We have already dealt with spin systems, particularly the example of a spin-½ particle. Our approach began with the empirical observation that a measurement of any spin component of a spin-½ particle could yield only the results +ħ/2 or −ħ/2. We introduced the basis states | z ±〉 for the two-dimensional Hilbert space ℋ. We also gave other basis states such as {| x ±〉} and {| y ±〉} in terms of the | z ±〉 states. From basis states and measurement values we constructed operators for the spin components S x , S y , and S z . With the operators in hand, we could then examine the algebraic relations between them (such as the commutation relation in Exercise 3.56). Our job here is to generalize our analysis to systems of arbitrary spin. To do this, we will reverse our chain of logic.
Abstract This chapter presents orbital angular momentum as the generator or rotations in space. The operators which carry out rotations in space form a group, which is a symmetry group of the Hamiltonian when the interaction is spherically symmetric. In such a case, there are certain useful statements which we can make about the role of invariant and irreducible subspaces of the Hilbert space under the group operations and how to calculate matrix elements of tensor operators within such spaces, that is, the Wigner-Eckart theorem.
In this chapter, we develop the background needed to move our quantum mechanical formalism fully into three dimensions. We discuss angular momentum, which is the motional complement to spin angular momentum, that we developed earlier. There are some critical differences between orbital and spin angular momentum, and we need to carefully discuss how they relate and how they differ. Why is angular momentum so important? Because in problems where we have central forces (forces that depend only on the distance between two particles and act along the line connecting the two particles), the angular momentum is conserved. So, it becomes an integral part of how we solve the problem. The most important problem we have to solve, the hydrogen atom, is a central force problem; hence, we need this work before we can move forward to some really interesting physics that will come in Chap. 11 . We cover the angular momentum operators, eigenvalues, and eigenstates, their matrix representations, how to combine angular momentum, and spherical harmonics. We revisit the EPR experiment using our new formalism. We also discuss some unique representations of angular momentum by mapping to two-dimensional harmonic oscillators, as developed by Schwinger.
Since the discovery a century ago, spin describing the intrinsic angular momentum of massive elementary particles has exposed its nature and significant roles in wide ranges of (relativistic) quantum phenomena and practical applications for future quantum technology. Emerging inconsistencies have also disclosed its telltale incomplete description. Finding relativistic spins (operators) of massive particles is a long-standing fundamental problem from the beginning of relativistic quantum mechanics. Here we present the rigorous derivation and the representation of spin operators from the spacetime symmetry. The covariant parity operation, defined by the spin operators, naturally leads to a fundamental equation equivalent to the covariant Dirac equation, which manifests existent relativistic spins. Proper understanding position operator in the Dirac theory on account of the spin operator through total angular momentum predicts no Zitterbewegung as well as conserving orbital and spin currents. The spin operators can be applicable for unraveling the inconsistencies and for exploring unveiled physics of massive particles.
The angular momentum plays an important role in quantum mechanics. It is no longer an ordinary vector, instead it is a vector operator, whose three components do not commute with each other. In quantum mechanics, the angular momentum becomes a vector operator. A general angular momentum is defined as a vector operator whose components are Hermitian operators satisfying the commutation relations similar to those for the orbital angular momentum. In order to obtain the eigenvalues and eigenfunctions, it is more convenient to express the orbital angular momentum operators in spherical polar coordinates. Spin angular momentum is an intrinsic property of elementary particles. In quantum computing, a qubit or quantum bit is a quantum version of the classical binary bit physically realized in a two-state quantum-mechanical system.